In prior art constant envelope transmission methods, such as Interplex modulation, Coherent Adaptive Subcarrier Modulation (CASM), majority-vote modulation, or inter-vote modulation, the best-case power efficiency is dependent on the relative phase relationships between component signals and equality or disparity in component signal power levels. As discussed below, it would be helpful to be able to improve the efficiency of constant envelope transmission methods.
In mathematics, the general field of optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize (or maximize) a function known in the art as “an objective function,” sometimes called a “cost function,” f(x), subject to a set of inequality and/or equality constraint equations, which can be expressed asgj(x)≦0 (Inequality Constraints)hj(x)=0 (Equality Constraints)for the jth constraint of M total constraint equations. S. S. Rao, “Optimization theory and Applications,” December, 1977.
There are a large number of sub-disciplines in mathematical programming including linear programming, nonlinear programming, quadratic programming, convex programming, integer programming, etc., depending on the form of the objective function. Mordecai Avriel (2003), Nonlinear Programming: Analysis and Methods, Dover Publishing (ISBN 0-486-43227-0). Other subdisciplines known in the art describe techniques that optimize the objective function over time, such as dynamic programming or optimal control theory. Techniques within these fields have not previously been applied to the generation of a constant envelope modulation. For example, nonlinear programming is a field of optimization techniques in which the objective function or the constraints or both contain nonlinear parts. One example of an optimization method from nonlinear programming is known as the penalty function method. In this method, an unconstrained minimization (i.e., in which the constraint is introduced indirectly) is used to solve a constrained minimization problem (i.e., where the constraint is introduced directly) by introducing a penalty factor, μk>0, and a penalty function p(x)>0 such that a penalized objective function is minimized iteratively (e.g., for penalty factor μk=1 in the first iteration, 5 in the 2nd iteration, 10 in the 3rd iteration) so that the “penalty” for violating the constraint grows after each iteration. By the end of the process, the minimization is conducted with a large penalty if the constraint is violated; consequently, the result approaches the true minimum for introducing the constraints directly. David G. Luenberger (1973), Introduction to Linear and Nonlinear Programming, Addison-Wesley Publishing Company. The form of the objective function, including the penalty functions and penalty factors, is given below.
      F    ⁡          (      x      )        =            f      ⁡              (        x        )              +                  μ        k            ⁢                        ∑                      j            =            1                    m                ⁢                              G            j                    ⁡                      [                                          g                j                            ⁡                              (                x                )                                      ]                              Gj(x) the penalty function and is some function of the j'th constraintμk is the penalty parameter corresponding to the kth iterationIt should be noted that there are many variations of the penalty function including absolute value penalty functions, quadratic loss penalty functions, etc. Just as there is no single equation that describes all of the sub-disciplines and methods in the theory of mathematical programming (optimization), there is also no single penalty function. These are all variations of solving the problem of minimizing an objective function subject to constraints.
In order to maximize efficiency of a nonlinear high-power amplifier, it is preferable to operate such amplifiers at or near saturation of their nonlinear region. Unfortunately, such operation leads to amplitude modulation to amplitude modulation (AM/AM) and amplitude modulation to phase modulation (AM/PM) distortions when the amplitude or envelope of the composite signal is not constant.
To reduce such distortions in space and ground-based communication and navigation transmission systems employing nonlinear amplifiers, two approaches have been undertaken. The first approach is to operate the amplifier in a linear region through output power backoff. This approach mitigates the effect of AM/AM and AM/PM distortions due to operation in the nonlinear region but is undesirable because it results in a loss of nonlinear amplifier efficiency as a result of not operating the amplifier at its maximum output power level (i.e., at saturation).
The second approach is to operate the amplifier in saturation while minimizing the variation in amplitude of the composite signal, or AM because such variation produces undesirable AM/AM and AM/PM distortions when the signal is passed through a nonlinear amplifier (as is the case in a typical space-based communication or navigation system). Several approaches have been previously proposed to maintain a constant envelope of the composite signal.
These approaches have been applied to modulate signals in Global Navigation Satellite Systems (GNSS) including the Global Positioning System (GPS), and the European Galileo navigation satellite system. These approaches maintain a constant carrier envelope according to a specific signal multiplexing (combining) technique. The approaches include: Coherent Adaptive Subcarrier Modulation (CASM), U.S. Pat. No. 6,430,213 to Dafesh; Quadrature Product Subcarrier Modulation (QPSM), U.S. Pat. No. 7,120,198 to Dafesh et al.; Interplex modulation, U.S. Pat. No. 6,335,951 to Cangiani et al.; weighted majority vote, U.S. Pat. No. 7,035,245 to Orr et al., or hybrids such as the inter-vote technique U.S. Pat. No. 7,154,962 to Cangiani et al. See also: J. Spilker et al., “Code Multiplexing via Majority Logic for GPS Modernization” Proceedings of the Institute of Navigation (ION) GPS-98, Sep. 15-18, 1998; P. A. Dafesh, T. M. Nguyen and S. Lazar, “Coherent Adaptive Subcarrier Modulation (CASM) for GPS Modernization,” Proceedings of the ION National Technical Meeting, January, 1999; and P. A. Dafesh, “Quadrature Product Subcarrier Modulation,” Proceedings of the IEEE Aerospace Conference, March 1999.
For these previous approaches, their efficiency is critically dependent on maintaining certain carrier phases and there is little flexibility to optimize the phase between different signal components, while minimizing the output power (producing a best-case power efficiency) required to transmit all of the signal components and pre-specified power levels. Maintaining phase relationships between signals is a desirable feature to meet legacy system requirements, such as the 90 degree phase relationship between P(Y) and C/A codes in the GPS system or to minimize interference between signals that overlap in spectrum.
In the case of GPS or GNSS systems in general, the number of signals that must be simultaneously transmitted in the future has increased from 2 to at least 5. This has resulted in a need to efficiently transmit these signals, the preferred method of which has been to employ some type of combining method that maintains a constant envelope composite signal amplitude (to eliminate AM-to-AM and AM-to-PM distortions), as described above. This combining approach should provide a composite signal without deleteriously impacting the power efficiency, defined by the following equation as the ratio of the sum of the component signal powers divided by the power of the composite signal:
  η  =            (                                    ∑                          n              =              1                        N                    ⁢                                                                  corr                n                                                    2                                    A          2                    )        .  Here Pdn=|corrn|2 is the desired value of the n'th component signal power, as measured by a correlation receiver matched to the component signal and PT=A2 is the total power of the composite transmitted signal, where A is the envelope (amplitude) of the composite signal. Corrn is the expected complex correlation level for the n'th component signal of N signals in the composite signal.
Additional requirements, such as the desire to maintain certain signal phase relationships, allow for adaptive signal power levels; and the possibility of increasing the number of signals in the future has led to increased signal losses and potentially severe self interference effects resulting from inter-modulation products inherent in these prior-art combining (modulation) methods. The methods also become increasingly difficult to optimize as the number of signals increases and are not well suited to combining other than BPSK signals. For example, the desire to broadcast composite signal envelope including Code Division Multiple Access (CDMA) waveforms (composed of multiple orthogonal pseudorandom noise code signals) and Orthogonal Frequency Division Multiplexing (OFDM) waveforms (composed of multiple orthogonal component signals at different frequencies) using nonlinear high power amplifiers operating near saturation makes it desirable to develop an efficient modulation method that may be applied to the optimization of a wide range of signal types, signal levels, phase relationships, and number of signals. Such methods may be used to efficiently transmit signals for application to both terrestrial wireless and space-based communication and navigation systems.